Israeli troops found the remains on Tuesday of the first rocket to be fired from Egypt since the July 3 overthrow of the Islamist government there, a military official said.
Both Israelis and Egyptians reported hearing several explosions in the southern city of Eilat on Thursday, the day after President Mohammed Morsi was toppled from power in Egypt.
Israel detected no signs of any cross-border shooting, but found the remains of a rocket on Tuesday, an official said, speaking on condition of anonymity.
An Israeli military spokesman said the rocket remnant had been discovered in the hills north of Eilat, a resort city on the Red Sea that abuts Egypt to the west and Jordan to the east.
The rocket was the first since the latest bout of unrest in Egypt that has put Israel on edge in part because of an increase of Islamist militancy in the Sinai region since an uprising toppled autocratic president Hosni Mubarak in 2011.
In Egypt, a hardline Salafist Islamist group called Ansar Bayt al-Maqdes, whose name means "Followers of Jerusalem," issued a statement claiming to have fired rockets at Eilat, targeting fuel depots and residential areas.
Additional reporting by Shadia Nasralla in Cairo; Writing by Allyn Fisher-Ilan; Editing by Mark Heinrich